Estimates of the Solutions of a System of Quasi-linear PDEs. A Probabilistic Scheme

Abstract

We propose a probabilistic scheme to estimate the Hölder norm and the gradient of the solutions of a system of quasi-linear PDEs of parabolic type. Indeed, thanks to the theory of Forward Backward stochastic differential equations, we are able to give a stochastic representation of the solutions of such systems of PDEs. Making use of Krylov and Safonov estimates, we deduce a Hölder estimate of these solutions in the case of uniformly parabolic systems with measurable coefficients. Moreover, from a variant of the Malliavin–Bismut integration by parts formula, we establish under appropriate assumptions an estimate of the supremum norm of the gradient of these solutions.

Keywords: Forward-backward stochastic differential equation, gradient estimate, Hölder estimate, integration by parts, system of quasi-linear PDEs of parabolic type.